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Mirrors > Home > MPE Home > Th. List > clwwlk1loop | Structured version Visualization version GIF version |
Description: A closed walk of length 1 is a loop. See also clwlkl1loop 29639. (Contributed by AV, 24-Apr-2021.) |
Ref | Expression |
---|---|
clwwlk1loop | β’ ((π β (ClWWalksβπΊ) β§ (β―βπ) = 1) β {(πβ0), (πβ0)} β (EdgβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . 4 β’ (VtxβπΊ) = (VtxβπΊ) | |
2 | eqid 2725 | . . . 4 β’ (EdgβπΊ) = (EdgβπΊ) | |
3 | 1, 2 | isclwwlk 29836 | . . 3 β’ (π β (ClWWalksβπΊ) β ((π β Word (VtxβπΊ) β§ π β β ) β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ))) |
4 | lsw1 14547 | . . . . . . . . . . 11 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = 1) β (lastSβπ) = (πβ0)) | |
5 | 4 | preq1d 4739 | . . . . . . . . . 10 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = 1) β {(lastSβπ), (πβ0)} = {(πβ0), (πβ0)}) |
6 | 5 | eleq1d 2810 | . . . . . . . . 9 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = 1) β ({(lastSβπ), (πβ0)} β (EdgβπΊ) β {(πβ0), (πβ0)} β (EdgβπΊ))) |
7 | 6 | biimpd 228 | . . . . . . . 8 β’ ((π β Word (VtxβπΊ) β§ (β―βπ) = 1) β ({(lastSβπ), (πβ0)} β (EdgβπΊ) β {(πβ0), (πβ0)} β (EdgβπΊ))) |
8 | 7 | ex 411 | . . . . . . 7 β’ (π β Word (VtxβπΊ) β ((β―βπ) = 1 β ({(lastSβπ), (πβ0)} β (EdgβπΊ) β {(πβ0), (πβ0)} β (EdgβπΊ)))) |
9 | 8 | com23 86 | . . . . . 6 β’ (π β Word (VtxβπΊ) β ({(lastSβπ), (πβ0)} β (EdgβπΊ) β ((β―βπ) = 1 β {(πβ0), (πβ0)} β (EdgβπΊ)))) |
10 | 9 | adantr 479 | . . . . 5 β’ ((π β Word (VtxβπΊ) β§ π β β ) β ({(lastSβπ), (πβ0)} β (EdgβπΊ) β ((β―βπ) = 1 β {(πβ0), (πβ0)} β (EdgβπΊ)))) |
11 | 10 | imp 405 | . . . 4 β’ (((π β Word (VtxβπΊ) β§ π β β ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)) β ((β―βπ) = 1 β {(πβ0), (πβ0)} β (EdgβπΊ))) |
12 | 11 | 3adant2 1128 | . . 3 β’ (((π β Word (VtxβπΊ) β§ π β β ) β§ βπ β (0..^((β―βπ) β 1)){(πβπ), (πβ(π + 1))} β (EdgβπΊ) β§ {(lastSβπ), (πβ0)} β (EdgβπΊ)) β ((β―βπ) = 1 β {(πβ0), (πβ0)} β (EdgβπΊ))) |
13 | 3, 12 | sylbi 216 | . 2 β’ (π β (ClWWalksβπΊ) β ((β―βπ) = 1 β {(πβ0), (πβ0)} β (EdgβπΊ))) |
14 | 13 | imp 405 | 1 β’ ((π β (ClWWalksβπΊ) β§ (β―βπ) = 1) β {(πβ0), (πβ0)} β (EdgβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 β c0 4318 {cpr 4626 βcfv 6542 (class class class)co 7415 0cc0 11136 1c1 11137 + caddc 11139 β cmin 11472 ..^cfzo 13657 β―chash 14319 Word cword 14494 lastSclsw 14542 Vtxcvtx 28851 Edgcedg 28902 ClWWalkscclwwlk 29833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-hash 14320 df-word 14495 df-lsw 14543 df-clwwlk 29834 |
This theorem is referenced by: umgrclwwlkge2 29843 |
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